Is an oath concerning items dedicated for the exclusive use of a slave binding regarding property designated for his service?

Primary text

If a person swears not to enter a house designated for a slave, ride his mount, or wear his clothing, and then enters the house designated for him, rides the designated mount, or wears the designated clothing, the oath is broken (Hanth). This interpretation holds because the reference in the oath is construed as designation (ikhtisas) rather than ownership (milkiyyah), as ownership is not applicable here. The designation implies what is under the slave's control or service.